ABSTRACT
Metricity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.3.2.2 Length of Shortest Path . . . . . . . . . . . . . . . . . . . 49 2.3.2.3 Real-Valued Cost . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.3.2.4 t-Cost Distance in Real Space . . . . . . . . . . . . . 50
2.3.3 Weighted Cost Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.3.4 Knight’s Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.4 Path-Dependent Neighborhoods and Distances . . . . . . . . . . . . . . . . . 52 2.4.1 Hyperoctagonal Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
2.4.1.1 Necessary and Sufficient Condition for Metricity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.4.2 Octagonal Distances in 2-D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.4.2.1 Best Simple Octagonal Distance . . . . . . . . . . . 58
2.4.3 Weighted t-Cost Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.5 Hyperspheres of Digital Distances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.5.1 Notions of Hyperspheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 2.5.2 Euclidean Hyperspheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.5.3 Hyperspheres of m-Neighbor Distance . . . . . . . . . . . . . . . . . . 62 2.5.3.1 Vertices of Hyperspheres . . . . . . . . . . . . . . . . . . . 65 2.5.3.2 Errors in Surface and Volume Estimations 65 2.5.3.3 Hyperspheres of Real m-Neighbor Distance 66
2.5.4 Hyperspheres of t-Cost Distance . . . . . . . . . . . . . . . . . . . . . . . . 68 2.5.5 Hyperspheres of Hyperoctagonal Distances . . . . . . . . . . . . . 69
2.5.5.1 Vertices of Hyperspheres and Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2.5.5.2 Special Cases of Hyperspheres in 2-D and 3-D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
2.5.6 Hyperspheres of Weighted t-Cost Distance . . . . . . . . . . . . . 75 2.5.6.1 Proximity to Euclidean Hyperspheres . . . . . 78
2.6 Error Estimation and Approximation of Euclidean Distance . . . 80 2.6.1 Notions of Error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 2.6.2 Error of m-Neighbor Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 2.6.3 Error of Real m-Neighbor Distance . . . . . . . . . . . . . . . . . . . . . 82 2.6.4 Error of t-Cost Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
2.6.4.1 Error of t-Cost Distance for Real Costs . . . 84 2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
Image processing in more than two dimensions has attracted a lot of interest recently. Three-dimensional image processing has several applications, such as computed tomography. The inclusion of time has increased the three spatial dimensions to four in studies involving moving objects. Various applications involving gray-scale pictures and objects with several features pertaining to a single dimension require representation in higher dimensions.
